Name resolution with Connection-specific DNS Suffix

I'm having a name resolution problem.

A cisco router configured by the home office serves as a DHCP server on this
small network. Part of the DHCP information is a domain name. Let's say
it's domain.net. This domain is a real domain name...a web page at
www.domain.net...and mail server at mail.domain.net, etc. It has one other
DNS entry on the name server. *.domain.net resolves to some IP address. So
everything not given a specific entry in DNS will resolve to that IP
address. Hopefully all of that makes sense.

Anyway...on the local network, there is no server. There are two Windows
Vista Business computers. One of them has a problem when it tries to
connect to another computer on the local network. When you ping any
unqualified name...it attaches the domain.net to the end of it and resolves
it to the IP address assigned on the name server. So this computer cannot
connect to any local machines by netbios name. You can ping the machines by
IP address.

For example:

Connection-specific DNS Suffix . : domain.net
Link-local IPv6 Address . . . . . : fe80::a8de:4737:ca79:5389%8
IPv4 Address. . . . . . . . . . . : 192.168.16.104
Subnet Mask . . . . . . . . . . . : 255.255.255.0
Default Gateway . . . . . . . . . : 192.168.16.1

If I ping the workstation name:

C:\Users\Gary>ping insp1420

Pinging insp1420.domain.net [66.66.666.66] with 32 bytes of data:
Reply from 66.66.666.66: bytes=32 time=66ms TTL=52
Reply from 66.66.666.66: bytes=32 time=36ms TTL=52
Reply from 66.66.666.66: bytes=32 time=36ms TTL=52
Reply from 66.66.666.66: bytes=32 time=35ms TTL=52

Notice how it automatically adds the domain.net suffix to the name I'm
pinging. I kind of fabricated the above ping...but this is the gist of it.
The ping will actually ping, because the IP address is valid....however it
doesn't resolve to the local IP of the workstation I'm actually trying to
ping because of that Connection-specific DNS Suffix.

Can anyone help me get this workstation to ping the local machines
correctly? It will ping any IP addresses correctly when I explicitly ping
the IP address. It will ping any external fully qualified domain name
correctly. These are stand alone workstations with Vista Business. The
router is configured by the home office for the specific use of some
software from remote locations and probably cannot be modified. I do intend
on putting a little netgear in it's place temporarily to see if the problem
goes away. But ultimately, I will have to contend with the Cisco router as
configured.

Thank you for any help.